
# Autarky

### Assumptions:

• Two goods: manufactured goods and $$👚$$ agricultural goods $$🌽$$
• Consumer needs to have non-zero food, but can have zero manufactured goods and still survive.
• $U=\ln (🌽) + \gamma \ln (👚+1)$
• Both goods are produced with CRS technology.
• $Y_👚 = z_👚 N_👚$ $Y_🌽 = z_🌽 N_🌽$
• No leisure decisions, I guess, just to keep things easy. Set price of 🌽 as numeraire?

### Formal Definition of Equilibrium:

Click to expand definition
###### Consumer's Problem:
Taking prices $$\{w,p\}$$ as given, chooses $$\{👚,🌽\}$$ to solve $\max_{👚,🌽} \ln (🌽) + \gamma \ln (👚+1)$ subject to the constraints that $👚,🌽 \geq 0$ $p👚 + 🌽 \leq w + \pi_👚 + \pi_🌽$
###### Manufacturing Firm's Problem:
Taking prices $$\{w,p\}$$ as given, chooses $$\{N_👚\}$$ to solve $\max p z_👚 N_👚 - w N_👚$
###### Agricultural Firm's Problem:
Taking prices $$\{w,p\}$$ as given, chooses $$\{N_🌽\}$$ to solve $\max z_🌽 N_🌽 - w N_🌽$
###### Market Clearing:
$N_👚 + N_🌽 = 1$ $👚 = z_👚 N_👚$ $🌽 = z_🌽 N_🌽$

### Solution to firms' problems:

For non-zero finite output, it must be that $p z_👚 = w = z_🌽$ and for zero output of 👚, it must be that $w \geq p z_👚$ In either case, profits will be zero.

### Solution to consumer's problem:

In an interior solution where $$👚>0$$, $\frac{(👚+1)}{\gamma 🌽} = p$ $p👚 + 🌽 = w$ $🌽 = \frac{w+p}{\left(1+\gamma\right)}$ $👚 = \frac{w\gamma-p}{p\left(1+\gamma\right)}$ And in a solution where $$👚=0$$, $🌽 = w$ Note that interior solution requires that $$p < w\gamma$$

Click to expand work for consumer problem Assume profits are zero. Will be true in equilibrium, so no big deal. Then Assuming interior solution where $$🌽,👚>0$$, Lagrangian is: $\mathcal{L} = \ln (🌽) + \gamma \ln (👚+1) - \lambda [ p👚 + 🌽 - w ]$ FOCs: ${1 \over 🌽} = \lambda$ ${\gamma \over (👚+1)} = \lambda p$ $p👚 + 🌽 = w$ Implies ${\gamma \over (👚+1)} = {1 \over 🌽} p$ Solve this to express each variable in terms of the other: $🌽=👚\frac{p}{\gamma}+\frac{p}{\gamma}$ $👚=🌽\frac{\gamma}{p}-1$ Combine these with budget constraint to solve explicitly for allocation variables: $🌽 = \frac{w+p}{\left(1+\gamma\right)}$ $👚 = \frac{w\gamma-p}{p\left(1+\gamma\right)}$

### Combine problems to get equilibrium

If $$\gamma z_👚 > 1$$, then we have the interior solution: $🌽 = z_🌽 \cdot \left( \frac{1+z_👚}{z_👚\left(1+\gamma\right)} \right)$ $👚 = \frac{\gamma z_👚 - 1}{\left(1+\gamma\right)}$ $N_🌽 = \frac{🌽}{z_{🌽}} = \frac{\left(z_{👚}+1\right)}{z_{👚}\left(1+\gamma\right)}$ $N_{👚} = \frac{👚}{z_{👚}}=\frac{\gamma-\frac{1}{z_{👚}}}{1+\gamma}$ Otherwise, if $$\gamma z_👚 \leq 1$$, $N_🌽 = 1$ $N_👚 = 0$ $🌽 = w = z_🌽$

Click to expand work for solving for the equilibrium allocations For an interior solution, we have the following characterizing equations:
From consumer: $\frac{(👚+1)}{\gamma 🌽} = p$ $p👚 + 🌽 = w$ $🌽 = \frac{w+p}{\left(1+\gamma\right)}$ $👚 = \frac{w\gamma-p}{p\left(1+\gamma\right)}$ $🌽,👚>0$ From firms: $p z_👚 = w = z_🌽$ From Market Clearing $N_👚 + N_🌽 = 1$ $Y_👚 = z_👚 N_👚$ $Y_🌽 = z_🌽 N_🌽$
Note how this implies that $p=\frac{ z_🌽 }{ z_👚 }$ And substitute out the prices to get allocations: $🌽 = \frac{ z_🌽 + {z_🌽 \over z_👚} }{\left(1+\gamma\right)} = z_🌽 \frac{1+z_👚}{z_👚\left(1+\gamma\right)}$ $👚 = \frac{z_🌽 \gamma-{z_🌽 \over z_👚}}{{z_🌽 \over z_👚}\left(1+\gamma\right)} = \frac{\gamma z_👚 - 1}{\left(1+\gamma\right)}$ $N_🌽 = \frac{🌽}{z_{🌽}} = \frac{\left(z_{👚}+1\right)}{z_{👚}\left(1+\gamma\right)}$ $N_{👚} = \frac{👚}{z_{👚}}=\frac{\gamma-\frac{1}{z_{👚}}}{1+\gamma}$ Note that the above equations are consistent with $$👚>0$$ iff $$\gamma z_👚 > 1$$. If this isn't the case, then it must instead be that $$👚=0$$ and $N_🌽 = 1$ $N_👚 = 0$ $🌽 = w = z_🌽$